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                <term xml:lang="en">resistance to antibiotics</term>
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              <p>Horizontal gene Transfer (HT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HT is believed to be the main instrument of developing the antibiotic resistance. In this work, we consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integrod-ifferential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal of this paper is to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular.</p>
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              <p>Le transfert horizontal de gènes (HT) est la transmission de matériel génétique entre deux organismes vivants, contrairement à la transmission verticale  qui désigne le transfert d'ADN d'un parent à sa progéniture. Il est prouvé que ce phénomène joue un rôle important dans l'évolution de certaines bactéries, notamment pour le développement d'une résistance aux antibiotiques. Nous considérons ici un processus stochastique à saut individu-centré, et l'équation intégro-differentielle non linéaire obtenue comme limite pour une population de grande taille. En supposant que les mutations sont petites, après un changement de variable et un passage à la limite, nous obtenons une équation d'Hamilton Jacobi. L'objectif de ce travail est de comparer ces differents modèles à l'aide de simulations numériques, et de déterminer si l'équation d'Hamilton-Jacobi parvient à capturer les phénomènes qualitatifs du modèle stochastique, notamment le sauvetage évolutif.</p>
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